Literature Reviews - Segmentation of Hand Bone for Bone Age Assessment - page 36

embodies 'elasticity of contour' and the 'preferred features in image' are deter-
mined by the functional as follows:

(2.9)

E [V ( S ) ] = E INT [V ( S ) ] + E EXT [V ( S ) ]

This functional of Eq. ( 2.9 ) represents the contour energy or the curve spline
bending energy. The task is to find the contour function, v ( s ) that minimizes the
functional E ( v ( s )) . The first term of Eq. ( 2.9 ) represents the internal energy:

∂ 2 V ( S )
∂ S 2

E INT [V ( S ) ] = ∫
0

∂ V ( S )
∂ S

(2.10)

W 1 ( S ) + W 2 ( S ) DS

This internal energy characterizes the deformation of the elastic contour at any
point s where w 1 ( s ) denotes the weight function that manipulates contour tension
and w 2 ( s ) denotes the weight function that manipulates contour rigidity. These ten-
sion and rigidity control the extent to which the contour can deform: Increasing
tension tends to reduce the length of the contour and hence the contour can get rid
of irrelevant ripples. Increasing rigidity tends to reduce the flexibility and hence
the contour can be smoother.
The second term of Eq. ( 2.9 ) represents the external energy that relates the con-
tour to the image plane:

E ext [ v ( s ) ] = ∫
0

(2.11)

I ( v ( s )) ds

The external energy evaluates the matching between the targeted object's
boundaries and the contour. This evaluation is based on certain feature image of
interest such as gradient image. If the v(s) located on the edges, then the external
energy in Eq. ( 2.11 ) at the point s is low. Hence, it is obvious that the minimum
external energy occurs when all the points s are located exactly on the detected
edges.
In short, the internal energy controls the behavior of curve itself whereas the
external energy determines how well the curve matches the features of inter-
est; both of them counterbalance each other. The minimum of the summation
of both energies produces a smooth contour that matches the feature of inter-
est, most commonly refers to edges of targeted object and thereby performs the
segmentation.
To apply active contour model in segmentation, first, establish the initial loca-
tion of point s in image planes adjacent to targeted object. These points collect
'evidence' locally in their territories and feedback to the contour energy. Next,
search the update of each point using local information by solving the Euler-
Lagrange equation when the contour is in equilibrium according to calculus of
variation. Conventionally, numerical algorithm is applied to solve the equation in
discrete approximation framework. Lastly, these steps repeat until stopping criteria
has been achieved.